Cavity-beam interaction and Longitudinal beam dynamics for CEPC DR&APDR 宫殿君 20160428

Outline CEPC DR Z-pole fundamental mode instabilities DR Higgs & Z phase shift CEPC APDR phase shift & longitudinal beam dynamics

Introduction Large high current storage rings suffer from two particular beam loading effects not present in smaller storage rings: First, the presence of phase modulations of the bunches due to gaps in the bunch train; Second, the excitation of fast growing longitudinal coupled bunch instabilities (CBI) by the detuned fundamental RF resonance.

Fundamental Mode Instability of Z-pole Fundamental mode CBI of Z-pole due to large cavity bandwidth and detuning. For M bunches, there are M coupled-bunch modes, the phase shift between adjacent bunches for mode number n: n = 0, 1, 2, ... M-1 CBI growth rate : f p μ+ = (pM + μ) f 0 + f s f p μ− = [(p+1)M − μ] f 0 + f s We refer to the mode μ=M-n as the mode μ= -n

CEPC DR Z machine parameters for CBI growth time calculation Baseline High-Lumi Unit Beam current 40.8 466 mA Beam energy 45.5 GeV Radiation loss/turn 34 MeV Bunch number 2708 21666 Rad. damping time 0.466 0.446 s Syn. frequency 0.106 0.204 kHz Rf frequency 650 MHz cavity number 16 48 Syn. phase 47.2 75.9 deg Cavity bandwidth 0.6 5.3 Detuning frequency -0.32 -10.9 Revolution frequency 3 Harmonic number 216660 Cavity voltage 6.3 2.9 MV Unloaded Q 1E+10 Loaded Q 1.1E+6 1.2E+5 Coupling factor 9.23E+3 8.12E+4 R/Q 206 Ohm

(1) Fundamental Mode Instability of Z-pole_baseline Longitudinal coupling-impedance Cavity resonance frequency 𝑍 || (𝑓)= 1 𝛽 𝑅 𝑠ℎ 2 1+𝑖 𝑄 𝐿 ( 𝑓 𝑓 𝑟𝑒𝑠 − 𝑓 𝑟𝑒𝑠 𝑓 Cavity acceleration mode impedance and beam spectrum of CEPC Z_baseline mode Growth time of CBI due to acceleration mode of CEPC Z_baseline

(2) Fundamental Mode Instability of Z-pole_high-lumi Longitudinal coupling-impedance Cavity resonance frequency 𝑍 || (𝑓)= 1 𝛽 𝑅 𝑠ℎ 2 1+𝑖 𝑄 𝐿 ( 𝑓 𝑓 𝑟𝑒𝑠 − 𝑓 𝑟𝑒𝑠 𝑓 Optimal detuning Cavity acceleration mode impedance and beam spectrum of CEPC Z-pole mode Growth time of CBI due to acceleration mode of CEPC Z operation (10 modes to be damped)

In CEPC DR Z_baseline, which detuning frequency and cavity bandwidth are small, no Acc. Mode exceed the threshold, when the beam current reach the design current. In CEPC DR Z_high-lumi, when the beam current reaches the design current 466 mA, from the μ=-1 mode to the μ=-10 mode instability due to the Acc. Mode will be excited. We can use the mode by mode damper system with digital filters to suppress these modes, just like SuperKEKB damper system:

Damper system for fundamental mode instability Use the mode by mode damper system with digital filters to suppress (SuperKEKB) Damper system for SuperKEKB: Kouki Hirosawa. Development of a coupled bunch instability damper caused by the acceleration mode for SuperKEKB. PASJ2016 TPU012 Functional block diagram of digital filter for SuperKEKB

DR Higgs & Z phase shift H W Z_baseline Z_high lumi Beam current I [mA] 19.2 97.1 40.8 465.8 Rev.freq [kHz] 3 Acc.phase [deg] 37.3 36.4 47.2 75.9 Vc [MV] 6.3 4.3 2.9 Voltage decrease (3% gap) 2.65% 9.82% 2.82% 69.86% Voltage decrease (5% gap) 4.42% 16.37% 4.69% 116.44% Phase shift (3% gap) [deg] 2.51 9.48 2.20 41.27 Phase shift (5% gap) [deg] 4.18 15.81 3.67 68.79

CEPC APDR longitudinal beam dynamics (1) APDR(61km 4+4DR) RF Parameters Based on Wangdou_20161109_61km parameters APDR H-low power H-high lumi W Z main ring type 8 double rings Circumference (km) 61 revolution frequency(kHz) 4.92 bunch charge (nC) 32 13.6 9.6 bunch number/train N 18 27 100 275 bunch spacing (ns) 185 123.3 33.3 12.1 Gap time interval Tg(us) 22.08 average beam current（mA） 11.02 17.00 26.80 52.00 circulating current(mA) 172.97 259.53 408.41 793.39 SR Loss/turn(GV) 2.96 0.58 0.06 SR Power*2(MW) 65.24 100.66 31.09 6.34 RF voltage Vrf(GV) 3.51 3.48 0.70 0.12 Cavity cell number 2 rf frequency(Hz) 6.50E+08 R/Q(Ω） 213.00 Quality Factor Q0 @2K 2.E+10 Shut impedance R(Ω) 4.26E+12 Cavity number 480 128 24 Pulse power/beam（MW） 512.00 768.21 236.88 48.40 Pulse power/cavity（MW） 1.07 1.60 1.85 2.02 Input power/cavity(kW) 135.92 209.71 242.88 264.33 Stored energy/cavity（J) 61.47 60.42 34.38 28.74

The phase shift is dropped by 40% APDR H-low power H-high lumi W Z Loaded quality factor QL 2.35E+05 1.54E+05 7.59E+04 5.82E+04 coupling parameter β 8.50E+04 1.30E+05 2.64E+05 3.44E+05 Loaded impedanceRL(Ω) 2.51E+07 1.64E+07 8.08E+06 6.20E+06 filling time(s) 1.15E-04 7.55E-05 3.72E-05 2.85E-05 unloaded filling time(s) 9.79 detuning frequency(Hz) -8.19E+02 -1.24E+03 -2.58E+03 -5.49E+03 optimum detuning(Hz) -8.80E+02 -1.30E+03 -2.89E+03 -9.46E+03 half 3db bandwidth(Hz) 1.38E+03 2.11E+03 4.28E+03 5.58E+03 detuning angle(deg) 32.51 31.73 34.05 59.45 Wilson syn phase(deg) Cavity Voltage(MV) 7.31 7.25 5.47 5.00 Effective length(m) 0.46 Acc. Gradient(MV/m) 15.84 15.71 11.85 10.83 RF station 8 cavity number/module 6 4 3 total module number 80 32 module/station 10 1 Pc@4K(kW) 22.5 22.1 2.5 0.4 Loss factor(V/pC) 0.31 0.25 0.24 HOM Power/Cavity(kW) 0.22 0.27 0.43 1.55 Phase shift(deg) 3.17 4.90 9.62 13.28 Voltage decrease(%) 3.26 4.69 8.99 26.60 Compared to 20170918 parameters, for W and Z, the opt. detuning frequency is decreased by 60%, and cavity bandwidth is reduced to ½, both are benefit for suppressing fundamental instability. The phase shift is dropped by 40%

(2) RF energy acceptance 𝜂 𝑅𝐹 =| 𝜀 max 𝐸 0 |= 𝑈 0 𝜋 𝛼 𝑝 ℎ 𝐸 0 𝐹(𝑞) 𝐹(𝑞)=2( 𝑞 2 −1 −arccos( 1 𝑞 )) 𝜂 𝑅𝐹 = 𝑈 0 𝜋 𝛼 𝑝 ℎ 𝐸 0 （ 1 tan 2 𝜑 − 𝜋 2 −𝜑） APDR H-LP H-HL W (1109) W (0918) Z (1109) Z (0918) Phase Shift [deg] 3.17 4.90 9.62 16.7 13.28 24.6 Voltage decrease 3.26% 4.69% 8.99 18 26.60 35 𝜂 𝑅𝐹 [%] 2.37 2.23 1.39 1.8 1.17 1.1 𝜂 𝑅𝐹 ′ [%] 2.01 1.71 0.81 0.6 0.70 0.3

(3) Nature bunch length APDR H-LP H-HL W (1109) W (0918) Z (1109) Phase Shift [deg] 3.17 4.90 9.62 16.7 13.28 24.6 Voltage decrease 3.26% 4.69% 8.99 18 26.60 35 Nature 𝜎 𝑧 [mm] 2.7 3.23 2.95 3.9 3.78 Nature 𝜎 𝑧 ′ [mm] 2.87 3.0 3.97 5.88 4.97 9.04

Luminosity decrease [%] Phase shift increases the bunch length, which decreases the Luminosity. APDR H-LP H-HL W (1109) W (0918) Z (1109) Z (0918) Phase Shift [deg] 3.17 4.90 9.62 16.7 13.28 24.6 Voltage decrease 3.26% 4.69% 8.99 18 26.60 35 Luminosity decrease [%] 12 20 34 75 39 83 Beam lifetime The beam lifetime due to beamstrahlung effect: APDR H-LP H-HL Phase Shift [deg] 3.17 4.90 Voltage decrease 3.26% 4.69% 𝜏 𝐵𝑆 [min] 37 35.1 34.1

momentum compaction(10^-5) (5) Longitudinal tune Some parameters about longitudinal motion APDR H-LP H-HL W E momentum compaction(10^-5) 1.48 3.10 Syn. frequency 𝑓 𝑠 (kHz) 0.34 0.19 0.30 0.28 0.13 0.12 synchrotron tune 0.070 0.069 0.039 0.038 0.062 0.056 0.026 0.024 Rad. Damping time (ms) 8.24 28.03 151.61 SR damping rate (s^-1) 121.36 35.67 6.60 Longitudinal tune is decreased, frequency spread between bunches makes Landau Damping stronger, bunches instability is easier to be damped.

(6) Dynamic aperture H-HL： 4.69% voltage drop 4.9 degrees phase shift： No phase shift：

Index CEPC DR 20170426 parameters CEPC APDR_61km 20160918&20161109 parameters

Parameters for CEPC double ring （wangdou20170426-100km_2mmy）   Pre-CDR Higgs W Z Number of IPs 2 Energy (GeV) 120 80 45.5 Circumference (km) 54 100 SR loss/turn (GeV) 3.1 1.67 0.33 0.034 Half crossing angle (mrad) 16.5 Piwinski angle 3.19 5.69 4.29 11.77 Ne/bunch (1011) 3.79 0.968 0.365 0.455 0.307 Bunch number 50 412 5534 21300 2770 Beam current (mA) 16.6 19.2 97.1 465.8 40.8 SR power /beam (MW) 51.7 32 16.1 1.4 Bending radius (km) 6.1 11 Momentum compaction (10-5) 3.4 1.14 4.49 IP x/y (m) 0.8/0.0012 0.171/0.002 0.171 /0.002 0.16/0.002 Emittance x/y (nm) 6.12/0.018 1.31/0.004 0.57/0.0017 1.48/0.0078 0.18/0.0037 Transverse IP (um) 69.97/0.15 15.0/0.089 9.9/0.059 15.4/0.125 5.6/0.086 x/y/IP 0.118/0.083 0.013/0.083 0.0055/0.062 0.008/0.054 0.006/0.054 RF Phase (degree) 153.0 128 126.9 165.3 136.2 VRF (GV) 6.87 2.1 0.41 0.14 0.05 f RF (MHz) (harmonic) 650 650 (217800) Nature z (mm) 2.14 2.72 3.37 3.97 3.83 Total z (mm) 2.65 2.9 4.0 HOM power/cavity (kw) 3.6 (5cell) 0.41(2cell) 0.36(2cell) 1.99(2cell) 0.12(2cell) Energy spread (%) 0.13 0.098 0.065 0.037 Energy acceptance (%) 1.5 Energy acceptance by RF (%) 6 1.1 0.68 n 0.23 0.26 0.15 0.12 0.22 Life time due to beamstrahlung_cal (minute) 47 52 F (hour glass) 0.96 0.98 0.99 Lmax/IP (1034cm-2s-1) 2.04 2.0 5.15 11.9

Parameter for CEPC partial double ring （wangdou20160918-61km）   New-61km H-high lumi. H-low power W Z Number of IPs 2 Energy (GeV) 120 80 45.5 Circumference (km) 61 SR loss/turn (GeV) 3.0 2.96 0.58 0.061 Half crossing angle (mrad) 15 Piwinski angle 1.88 1.84 5.2 6.4 Ne/bunch (1011) 3.91 2.0 1.98 1.16 0.78 Bunch number 54 107 70 400 1100 Beam current (mA) 16.6 16.9 11.0 36.5 67.6 SR power /beam (MW) 50 32.5 21.3 4.1 Bending radius (km) 6.1 6.2 Momentum compaction (10-5) 3.25 1.48 1.44 2.9 IP x/y (m) 0.43/0.00105 0.272/0.0013 0.275 /0.0013 0.1/0.001 Emittance x/y (nm) 6.28/0.04 2.05/0.0062 2.05 /0.0062 0.93/0.0078 0.88/0.008 Transverse IP (um) 51.7/0.2 23.7/0.09 9.7/0.088 9.4/0.089 x/IP 0.118 0.041 0.042 0.013 0.01 y/IP 0.074 0.11 0.073 0.072 VRF (GV) 6.99 3.48 3.51 0.74 f RF (MHz) 650 Nature z (mm) 2.19 2.7 2.95 3.78 Total z (mm) 2.47 3.35 4.0 HOM power/cavity (kw) 3.9 0.48 0.88 0.99 Energy spread (%) 0.13 0.087 0.05 Energy acceptance (%) 2.1 Energy acceptance by RF (%) 6 2.3 2.4 1.7 1.2 n 0.31 0.35 0.34 0.49 Lmax/IP (1034cm-2s-1) 2.02 3.1 2.01 4.3 4.48

Parameter for CEPC partial double ring （wangdou20161109-61km）   Pre-CDR H-high lumi. H-low power W Z Z-5cell Number of IPs 2 Energy (GeV) 120 80 45.5 Circumference (km) 54 61 SR loss/turn (GeV) 3.1 2.96 0.58 0.061 Half crossing angle (mrad) 15 Piwinski angle 1.88 1.84 4.11 5.86 5.87 Ne/bunch (1011) 3.79 2.0 1.98 0.85 0.6 Bunch number 50 107 70 400 1100 700 Beam current (mA) 16.6 16.9 11.0 26.8 52.0 33.1 SR power /beam (MW) 51.7 32.5 15.7 3.2 Bending radius (km) 6.1 6.2 Momentum compaction (10-5) 3.4 1.48 IP x/y (m) 0.8/0.0012 0.272/0.0013 0.275 /0.0013 0.16/0.001 0.12/0.001 Emittance x/y (nm) 6.12/0.018 2.05/0.0062 2.05 /0.0062 0.93/0.003 0.87/0.0046 Transverse IP (um) 69.97/0.15 23.7/0.09 12.2/0.056 10.2/0.068 x/IP 0.118 0.041 0.042 0.0145 0.0098 y/IP 0.083 0.11 0.084 0.073 VRF (GV) 6.87 3.48 3.51 0.7 0.12 f RF (MHz) 650 Nature z (mm) 2.14 2.7 3.23 3.9 Total z (mm) 2.65 2.95 2.9 3.35 4.0 HOM power/cavity (kw) 3.6 0.74 0.48 0.47 0.59 0.93 Energy spread (%) 0.13 0.087 0.05 Energy acceptance (%) Energy acceptance by RF (%) 6 2.3 2.4 1.3 1.1 n 0.23 0.35 0.34 0.28 0.24 Life time due to beamstrahlung_cal (minute) 47 37 F (hour glass) 0.68 0.82 0.89 0.92 Lmax/IP (1034cm-2s-1) 2.04 2.01 3.5 3.44 2.2